Measurement and modeling of density and viscosity of the NaClO4 + H2O + poly(ethylene glycol) system at various temperatures

Fluid Phase Equilibria 334 (2012) 22–29

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Measurement and modeling of density and viscosity of the NaClO4 + H2 O + poly(ethylene glycol) system at various temperatures Yecid P. Jimenez a , María E. Taboada a,b , Teófilo A. Graber a,b , Héctor R. Galleguillos a,b,∗ a b

Department of Chemical and Mineral Process Engineering, University of Antofagasta, Av. Angamos 601, Antofagasta, Chile Centro de Investigación Científico y Tecnológico para la Minería (CICITEM), Av. José Miguel Carrera 1701, 4◦ piso, Antofagasta, Chile

a r t i c l e

i n f o

Article history: Received 4 April 2012 Received in revised form 13 July 2012 Accepted 19 July 2012 Available online 27 July 2012 Keywords: Density Viscosity Sodium perchlorate, Poly(ethylene glycol), NRTL

a b s t r a c t The densities and viscosities of unsaturated solutions of sodium perchlorate + poly(ethylene glycol) + water were experimentally measured at temperatures of 288.15, 298.15 and 308.15 K. The solution concentrations of poly(ethylene glycol) with an average molecular weight of 4000 ranged from 0.25 to 33.91 (mass%) and from 10.12 to 37 (mass%) of NaClO4 . The electrolyte and polymer non-random two liquid models were extended for representing of the excess molar volume and the dynamic viscosity of the above system. At these specific temperatures, the model correlations compared favorably with the experimental data for both the density and the viscosity. © 2012 Elsevier B.V. All rights reserved.

1. Introduction During the last few years, numerous studies have been carried out on aqueous two-phase systems (ATPSs) containing poly(ethylene glycol) (PEG). ATPSs have numerous uses in biotechnology [1,2], chemical partitioning [3–5] and the extractive crystallization of inorganic salts [6]. Both thermodynamic and transport properties of aqueous polymer-salt solutions are very important in chemical engineering and in other disciplines that contribute to the understanding the fundamentals of separation processes, fluid transport, wastewater treatment, etc. Densities and viscosities of aqueous solutions are useful for pipe sizing, pump calculations, material balance calculations, design of crystallizers, evaporators, etc. In addition, they are useful in determining the nature of solute–solvent and solute–solute interactions. While density and viscosity data of aqueous PEG + salt systems are available, correlation and prediction models with a thermodynamic basis are scarce. Snyder et al. [7] measured the viscosity and density of each phase of the ATPSs formed by PEG and magnesium sulfate, sodium sulfate, sodium carbonate, ammonium sulfate, and potassium phosphate at 298.15 K. Other comparable work is reported by Mei et al. [8]. Zafarani-Moattar et al. [9] measured the densities of aqueous solutions for several PEG + salt systems at 298.15, 308.15 and

∗ Corresponding author at: Department of Chemical and Mineral Process Engineering, University of Antofagasta, Av. Angamos 601, Antofagasta, Chile. Tel.: +56 55 637344; fax: +56 55 637801. E-mail address: [email protected] (H.R. Galleguillos). 0378-3812/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2012.07.017

318.15 K. The salts of sodium sulfate, sodium carbonate, ammonium sulfate, and a mixture of dipotassium hydrogen phosphate and potassium dihydrogen phosphate were used in these experiments. Similar work was also reported by Zafarani-Moattar and Mehrdad [10]. Graber et al. [11] measured the refractive index, density, and viscosity of unsaturated solutions of a sodium nitrate + PEG 4000 + water system at six temperatures ranging from 288.15 to 313.15 K and correlated them using Othmer’s rule. Other similar works were also reported by Graber et al. [12]. Telis-Romero et al. [13] reported dynamic viscosities of a potassium phosphate + PEG 1500 + water system at 303.15 K for a single- and two-phase region; for the single-phase systems, the Grunger-like equation was used, and for the two-phase region, an empirical equation was developed. In both cases, the average absolute deviation was low. Gonc¸alves et al. [14] measured and reported the kinematic viscosities of salt (sodium dihydrogen phosphate, sodium hydrogen phosphate, dipotassium hydrogen phosphate or potassium dihydrogen phosphate) + PEG 1000 + water ternary systems at 298.2 K. A reformulated Kumar equation was employed to correlate and predict the kinematic viscosities. Additionally, similar work was reported by Taboada et al. [15], Zafarani-Moattar and Hamzehzadeh [16], Murugesan and Perumalsamy [17,18] and Regupathi et al. [19]. Finally, Sadeghi et al. [20] reported density data from a sodium tungstate + PEG 6000 (mass fraction of 0.02) + water system for temperatures from 288.15 to 308.15 K. The Pitzer model was used to correlate the experimental apparent molar volume data at varying temperatures. In a previous article [21], we reported liquid–liquid equilibrium data for a sodium perchlorate + PEG 4000 + water system at

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29 Table 1 Supplier and purity of the chemicals used. Chemical

Supplier

Purity

Purification method

PEG 4000 NaClO4 ·H2 O

Merck Merck

Synthesis grade ≥0.99 (mass fraction)

None None

23

values lower than 32.67 mPa s the uncertainty was ±1.5 × 10−3 mPa s. Both density and viscosity were measured in triplicate, and their averages were calculated. 3. Thermodynamic framework

288.15, 298.15 and 308.15 K. This system, as previously mentioned, is of particular importance as the ion perchlorate is an impurity in the mineral nitrate deposits in northern Chile. Additionally, the possibility of partitioning the nitrate ion in this system or the perchlorate ion in a sodium nitrate + PEG 4000 + water system could lead to an alternative process for the attenuation of this impurity. To further our understanding of this system, the present work reports on the experimental data on density and viscosity for unsaturated solutions at 288.15, 298.15 and 308. 15 K. The experimentally determined values of density and viscosity were correlated with an extended NRTL model [22–25]. To begin the process of correlating the experimental data for the ternary system, we first correlated the literature data for the corresponding binary systems [8,10,11,26–32]. The Eyring equation, which relates the transport and thermodynamic properties, was used to correlate the viscosity data. Additionally, these properties measured for the present system have not previously been reported in the literature. 2. Experimental 2.1. Materials Poly(ethylene glycol) of synthesis grade (CAS Registry No. 25322-68-3, Batch S5348590) with an average molecular weight of 4000 g mol−1 (3500–4500 g mol−1 , melting point: 58.4–60.8 ◦ C) and sodium perchlorate monohydrate of pro analysis grade with a purity of ≥0.99 mass fraction (CAS Registry No. 7791-07-3, Batch A944064) were procured from Merck and used without further purification. This information is summarized in Table 1. Milli-Q quality distilled water (0.054 ␮S cm−1 ) was used in all of the experiments. 2.2. Apparatus and procedures The solutions were prepared by mass, using an analytical balance with a precision of ±0.07 mg (Mettler Toledo Co., model AX-204). The experimental error in the concentration was less than 0.01 wt%. Each solution was agitated using a Mini Vortexer (VWR Scientific Products Co.). The densities of the solutions were measured with a Mettler Toledo DE50 vibrating tube densimeter with an uncertainty of ±5 × 10−5 g cm−3 . The densimeter was calibrated at atmospheric pressure using air and distilled deionized water as a reference substance prior to the initiation of each run of measurements at a given temperature. The densimeter has a self-contained Peltier system for temperature control with an uncertainty of ±0.01 K. The time to reach a stable temperature was 300 s. The kinematic viscosities were measured with a calibrated micro-Ostwald viscometer. A Schott-Gerate automatic measuring unit (model AVS 310) equipped with a thermostat (Schott-Gerate, model CT 52) to regulate temperature to within ±0.02 K was used for these measurements. The absolute viscosity was obtained by multiplying the kinematic viscosity by the corresponding density. The uncertainty in viscosity measurements that ranged from 78.54 to 32.67 mPa s was ±4 × 10−2 mPa s. For viscosity

Correlation of the density and viscosity for the ternary system using the NRTL model requires the interaction parameters of the constituent binary systems. The following sections contain the equations for both the binary and ternary systems. 3.1. Correlation of density of binary aqueous solutions For the NaClO4 + H2 O system, the relationship between the density and the excess volume of a solution is given by the following equation: d=

1000(1 + mM2 ) (1/M1 )V10 + mV20 + V ex

(1)

where d is the density of solution (in g cm−3 ) and V10 , V20 and Vex are the molar volume of pure solvent, partial molar volume of salt at infinite dilution and the excess volume of the solution (the first two in cm3 mol−1 and the last in cm3 kg−1 ), respectively. The V10 values are 18.0162, 18.0522 and 18.108 cm3 mol−1 at 288.15, 298.15 and 308.15 K, respectively. M1 , M2 and m are the molar mass of the solvent, molar mass of the salt (both in kg mol−1 ) and molality of the solution (in mol kg−1 ), respectively. The partial molar volume at infinite dilution for each salt can also be regarded as an adjustable parameter. Its numerical value can be obtained by fitting the model to density data. In the present work, V20 is considered to be an adjustable parameter. The molar excess Gibbs energy and the excess volume are expressed as a sum of the long-range and short-range contributions: g ex∗ = g ex∗,LR + g ex∗,SR

(2)

V ex = V ex,LR + V ex,SR

(3)

where LR and SR refer to the long-range and short-range contributions, respectively. The Vex value is calculated by the Chen’s non-random two liquid (NRTL) model [22] derived by Humffray [33]. v v For the NRTL interaction parameters w,ca and ca,w , a temperature-dependent function is assumed as follows: ijv =

aijv RT

(4)

The temperature-dependent function used for partial molar volume of salt at infinite dilution is V20 = a + bT

(5)

where avij , a and b are adjustable parameters. For the PEG 4000 + H2 O system, Eq. (1) was also used to established the relationship between the density and the excess volume. In this case, d is the density of solution (in g cm−3 ), and M1 , M2 and m are the molar mass of the solvent, molar mass of the polymer (both in kg mol−1 ) and molality of the solution (in mol kg−1 ), respectively. V10 , V20 and Vex are the molar volume of pure solvent, molar volume of polymer at infinite dilution and the excess volume of the solution (the units are those mentioned above), respectively. The molar volume of PEG is calculated from the infinite dilution apparent molar volume of polymer calculated from the density data of aqueous solutions of PEG 4000 [10,11]. Finally, the Vex value is calculated by the NRTL polymer model [23] derived by Sadeghi et al.

24

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

[34] but expressed in the unsymmetrical convention, as follows: V ex,NRTL RT



= xw

 + rxp

v ((1 + ln G )(X + rX G ) − rX G rXs Gsw sw sw w s sw s sw ln Gsw )

(Xw + rXs Gsw )2 v ((1 + ln G )(rX + X G ) − X G Xw Gws ws ws s w ws w ws ln Gws )





(6)

xr

I i,I 

(7)

xr j J j,J

Gji = exp(−˛ji ji ) ji =

gji − gii RT

=

=−



 1 4A Ix xI

M1

I



ln(1 + 



Ix )

(14)

where

where the subscripts w and s stand for water and segment of polymer, respectively. Xi is the effective local mole fraction based on the number of moles of segments and solvent, and the other terms are defined by the following:

J

ex∗,PDH gLR

RT

(rXs + Xw Gws )2

v + G  v (1 + ln G )) −rxp (ws sw sw sw

Xi =

The Pitzer–Debye–Hückel (PDH) equation [36] normalized to mole fractions of unity for solvent and zero for electrolytes is used to calculate the gex*,LR . The expression has the following form [36]:

(8)

aji

(9)

RT

In the above equations, the species i and j can be solvent or segments. xI and ri,I are the mole fraction and number of segments species i in component I, respectively. gji and gii are energies of interaction between j − i and i − i pairs of species, respectively. ˛ is the non-randomness factor, and ijv represents pressure derivatives of  ij , (∂ ij /∂p)T,x . A value of r = 1 is used for water. For the polymer, the value of r is the ratio of the molar volume of polymer to that of water. R is the gas constant.

A =

1 3

 2N 1/2 e2 3/2 A Vs

(15)

εT KT

In the above expressions, A is the Debye–Hückel constant for the osmotic coefficient,  is the closest distance parameter (set to 14.9). NA , K and e are Avogadro’s number, Boltzmann’s constant and the electronic charge, respectively. Ix , is the ionic strength in mole  fraction basis, and Ix = 0.5 i xi Zi2 . Vs and εT are the molar volume and dielectric constant of the solvent, respectively. The extended NRTL model [24,25] for gex*,SR can be written as ex∗,NRTL gSR

RT = xw

2Xc Gca,w ca.w + rXs Gsw sw 2Xc Gca,s ca.s + Xw Gws ws + rxp 2Xc Gca,w + rXs Gsw + Xw 2Xc Gca,s + rXs + Xw Gws

+ Zc xc

rXs Gs,ca s.ca +Xw Gw,ca w,ca rXs Gs,ca s.ca +Xw Gw,ca w,ca +Za xa Xa +rXs Gs,ca +Xw Gw,ca Xc + rXs Gs,ca + Xw Gw,ca

− Zc xc ln c∞ − Za xa ln a∞ − xp ln p∞

(16)

3.2. Correlation of density of ternary aqueous solutions Eq. (1) may be generalized as follows: d=

1000(1 + mM2 + mM3 )

(10)

(1/M1 )V10 + mV20 + mV30 + V ex

where the subscripts 1, 2 and 3 stand for solvent, salt and polymer, respectively. The notation has the conventional meaning. For Eq. (10), the molar masses and molar volumes are in kg mol−1 and cm3 mol−1 , respectively, and the poly(ethylene glycol) is considered a solute. The molar excess Gibbs energy, gex* , considering the unsymmetrical reference state, is calculated as a sum of three contributions: g ex∗ = g ex∗,Comb + g ex∗,LR + g ex∗,SR gex*,LR

is the combinatorial contribution, is the where long-range interaction contribution and Gex*,SR is the short-range interaction contribution. The Flory–Huggins expression [35], the Pitzer–Debye–Hückel equation (PDH) [36] and the extended NRTL model [25] are used to calculate gex*,Comb , gex*,LR and gex*,SR , respectively. The Flory–Huggins equation [35] for gex*,Comb can be written as

I

ϕ   I xI

−

XI = ϕI KI

Gmm = exp(−˛mm mm )

xI ln I∞FH

(12)

I

amm (gmm − gm m ) = RT RT cm = am = ca,m

mm =



rx J J J

(18)

(13)

In these expressions, the species I and J can be water and polymer molecules or ions. A value of r = 1 is used for ions and water. For the polymer, the value of r is the ratio of the molar volume of polymer to that of water. I∞ is the infinite-dilution activity coefficient.

mc = ma = m,ca (19)

where gmm and gm m are the energy of interaction between the m − m and m − m species, respectively. These are inherently symmetric (gmm = gm m ). The species m and m can be solvent molecules, salt or segments. Appropriate differentiation of the (12), (14) and (16) equations yields the following expressions for excess volume:

rI xI

(17)

G and  are energy parameters and given by

where ϕI =

(KI = ZI for ions and unity for polymer and water)

(11)

gex*,Comb

 g ex∗,Comb = xI ln RT

In the above equations the subscripts w, s, p, ca, c and a refer to water, the polymer segment, polymer, salt, cation and anion, respectively. XI is the effective local mole fraction based on the number of moles of segments, ions and solvent and is expressed by the following:

ex,PDH VLR =



 1 Ix Av ln 1 +  Ix M1 

 (20)

where

Av = −4RT

∂A ∂p

(21) T,n

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

⎡ ex,NRTL VSR

RT

25

v (1 + ln G ) + 2X G v (rXs Gsw sw sw c ca,w ca,w (1 + ln Gca,w ))(rXs Gsw + 2Xc Gca,w + Xw )−

⎢ ⎢ (rX G  + 2X G  )(rX G (−˛ v ) + 2X G (−˛ v )) c ca,w ca,w s sw c ca,w ⎢ s sw sw sw ca,w = xw ⎢ 2 ⎢ (rXs Gsw + 2Xc Gca,w + Xw ) ⎣ ⎡

v (1 + ln G v (2Xc Gca,s ca,s ca,s ) + Xw Gws ws (1 + ln Gws ))(rXs + 2Xc Gca,s + Xw Gws )−

⎢ ⎢ (2X G  + X G  )(2X G (−˛ v ) + X G (−˛ v )) c ca,s ca,s w ws ws c ca,s w ws ⎢ ca,s ws rxp ⎢ ⎢ (rXs + 2Xc Gca,s + Xw Gws )2 ⎣ ⎡

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

v (1 + ln G v (rXs Gs,ca s,ca s,ca ) + Xw Gw,ca w,ca (1 + ln Gw,ca ))(rXs Gs,ca + Xa + Xw Gw,ca )−

⎢ ⎢ (rX G  + X G  )(rX G (−˛ v ) + X G (−˛ v )) w w,ca w,ca s s,ca w w,ca ⎢ s s,ca s,ca s,ca w,ca +Zc xc ⎢ ⎢ (rXs Gs,ca + Xa + Xw Gw,ca )2 ⎣ ⎡



⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

v (1 + ln G v rXs Gs,ca s,ca s,ca ) + Xw Gw,ca w,ca (1 + ln Gw,ca ) (rXs Gs,ca + Xc + Xw Gw,ca )−

⎢ ⎢ (rX G  + X G  )(rX G (−˛ v ) + X G (−˛ v )) w w,ca w,ca s s,ca w w,ca ⎢ s s,ca s,ca s,ca w,ca +Za xa ⎢ ⎢ (rXs Gs,ca + Xc + Xw Gw,ca )2 ⎣

(22)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

v + G  v (1 + ln G )) − Z x ( v v −rxp (ws sw sw sw c c w,ca + Gca,w ca,w (1 + ln Gca,w )) v v −Za xa (w,ca + Gca,w ca,w (1 + ln Gca,w ))

In the above expressions, Av is the Debye–Hückel constant for volume, and ijv represents pressure derivatives of  ij , (∂ ij /∂p)T,x . As observed from Eqs. (20) and (22), the Flory–Huggins equation does not appear in the relationship for the excess molar volume because the combinatorial term is independent of pressure or temperature. From Eq. (22) for the correlation of excess volume (or density), the model requires four new binary interv , and  v , for the segment–salt action parameters,  s,ca ,  ca,s , s,ca ca,s interaction pair and salt–segment interaction pair. The other interaction pairs are determined from the binary systems. In this work, these parameters were obtained by fitting the model to the density data. Finally, the expression for the excess volume of the ternary system can be expressed as in Eq. (3). 3.3. Correlation of viscosity of binary aqueous solutions For the NaClO4 + H2 O system, the relationship between viscosity and the molar excess Gibbs energy of activation for flow is established by the following equation [37]:

− w = A w

N solu  ci

exp

 g ex∗  RT

(23)

N=i

where and w are the solution viscosity and solvent viscosity (both Pa s), respectively, ci is the molar concentration of the solute species i, and gex* and Nsolu are the molar excess Gibbs energy of activation for flow and the total number of solute species, respectively. A (L mol−1 ) is taken as an empirically adjustable parameter. The molar excess Gibbs energy of an aqueous electrolyte solution is assumed to be a sum of the contributions of a long-range electrostatic interaction term, gex*,LR , and a short-range interaction term, gex*,SR , as in Eq. (2). The gex* value is calculated using Chen’s NRTL model [22] extended by Sadeghi [38]. In the case of the PEG 4000 + H2 O system, the relationship used to correlate the viscosity data is the same Eq. (23), the difference

being that the molar excess Gibbs energy is the sum of the combinatorial contribution gex*,comb (Eq. (12)) and a short-range interaction term, gex*,SR [23]. This last term can be expressed in unsymmetrical convention as ex∗,NRTL gSR

RT

= xw

rXs Gsw sw Xw Gws ws + rxp − xp ln p∞ rXs Gsw + Xw rXs + Xw Gws

(24)

3.4. Correlation of viscosity of ternary aqueous solutions Eq. (23) is generalized considering that the excess Gibbs energy is the sum of the three contributions mentioned above in Eq. (11). It is important to mention that for the ternary system, the polymer is assumed to be another solute. Eqs. (12), (14) and (16) are used to calculate the combinatorial, gex*Comb , long-range interaction, gex*,LR , and short-range interaction, gex*SR , contributions, respectively. The parameter A is assumed to be equal to A2 + A3 + A4 , where A2 and A3 are adjustable parameters obtained from the correlation of the viscosity values of the binary aqueous solutions. A4 is taken to be an empirically adjustable parameter of the ternary system. The model also requires two new binary interaction parameters,  s,ca and  ca,s , for the segment–salt and salt–segment interaction pairs because the other interaction pairs are determined from binary systems. These parameters were obtained from fitting the model to the viscosity data. 4. Results and discussion 4.1. Experimental results In this work, the experimental values of density and viscosity of the ternary system NaClO4 + PEG 4000 + water at three temperatures, 288.15, 298.15, and 308.15 K, were measured. The sodium perchlorate and poly(ethylene glycol) concentrations used in determining the density and the kinematic viscosity were selected by down of the binodal curve reported in a previous work [21]. These experimental concentrations were divided into four sets

26

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

Table 2 Experimental values for the density , viscosity in aqueous solutions of NaClO4 and PEG 4000 for different mass percent of sodium perchlorate (w1 ) and PEG (w2 ) at T = 288.15, 298.15 and 308.15 K.a w1 Set a 25.06 25.12 25.39 26.08 27.21 28.00 28.42 29.86 30.00 33.00 35.00 37.04 Set b 20.06 20.12 20.39 21.08 22.21 23.42 24.90 25.00 28.00 30.00 33.00 Set c 15.13 15.41 16.11 17.19 18.00 18.36 19.97 20.00 22.99 25.00 28.00 28.09 Set d 10.12 10.39 11.08 12.21 13.42 14.98 16.67 18.00 20.00 22.86 23.00

w2

/g cm−3 288.15 K

/mPa s

/g cm−3 298.15 K

/mPa s

/g cm−3 308.15 K

/mPa s

33.91 30.00 25.00 20.00 15.00 11.63 10.00 5.00 4.81 1.34 0.64 0.25

1.24993 1.24332 1.23690 1.23469 1.23523 1.23605 1.23693 1.24095 1.24187 1.26354 1.28161 1.30120

78.54 50.71 28.88 17.24 9.81 6.64 5.39 2.96 2.76 1.91 1.84 1.86

1.24226 1.23574 1.22949 1.22736 1.22778 1.22835 1.22967 1.23356 1.23439 1.25582 1.27363 1.29309

51.02 33.87 20.48 12.39 7.26 4.80 3.99 2.26 2.18 1.50 1.46 1.46

1.23464 1.22819 1.22184 1.21981 1.22023 1.22083 1.22211 1.22588 1.22650 1.24803 1.26561 1.28498

36.62 25.12 15.01 10.03 5.60 3.95 3.17 1.78 1.74 1.23 1.18 1.19

33.91 30.00 25.00 20.00 15.00 10.00 5.00 4.87 1.34 0.54 0.25

1.20725 1.20094 1.19417 1.19118 1.19155 1.19272 1.19651 1.19728 1.21716 1.23379 1.26101

62.46 41.06 24.75 14.40 8.56 4.83 2.60 2.58 1.69 1.62 1.67

1.20028 1.19389 1.18738 1.18450 1.18479 1.18617 1.18983 1.19067 1.21023 1.22655 1.25384

41.77 29.07 17.46 10.76 6.38 3.73 2.01 1.98 1.35 1.28 1.32

1.19315 1.18691 1.18034 1.17758 1.17769 1.17914 1.18284 1.18367 1.20286 1.21916 1.24618

29.91 20.58 13.19 8.16 4.97 2.84 1.62 1.61 1.11 1.04 1.09

30.00 25.00 20.00 15.00 11.22 10.00 5.00 4.89 1.32 0.54 0.26 0.25

1.16101 1.15412 1.15056 1.15020 1.14987 1.15069 1.15485 1.15492 1.17332 1.18908 1.21502 1.21585

35.00 21.35 13.09 7.76 5.20 4.49 2.42 2.40 1.59 1.48 1.49 1.51

1.15471 1.14797 1.14457 1.14432 1.14402 1.14475 1.14885 1.14901 1.16696 1.18252 1.20815 1.20892

24.48 15.48 9.51 5.85 3.91 3.36 1.89 1.88 1.25 1.18 1.18 1.18

1.14805 1.14146 1.13813 1.13805 1.13770 1.13847 1.14256 1.14275 1.16046 1.17577 1.20090 1.20157

18.59 11.88 7.42 4.66 3.04 2.69 1.53 1.53 1.02 0.96 0.98 0.98

30.00 25.00 20.00 15.00 10.00 4.99 2.49 1.21 0.58 0.25 0.22

1.12347 1.11625 1.11231 1.11174 1.11197 1.11511 1.12368 1.13196 1.14700 1.17037 1.17143

32.67 20.11 12.28 7.43 4.21 2.32 1.73 1.48 1.39 1.39 1.37

1.11756 1.11069 1.10694 1.10651 1.10678 1.10997 1.11829 1.12645 1.14088 1.16365 1.16475

22.27 14.47 9.07 5.59 3.30 1.81 1.36 1.16 1.10 1.10 1.08

1.11138 1.10475 1.10112 1.10082 1.10124 1.10437 1.11267 1.12069 1.13516 1.15754 1.15865

16.51 10.84 6.87 4.32 2.52 1.46 1.10 0.96 0.91 0.90 0.90

Fig. 1. Experimental concentrations prepared: ×, set a; , set b; , set c; +, set d (cf. Table 2) and binodal curve, −− [21].

4.2. Correlation of density data for binary systems The density values of a NaClO4 + H2 O binary system in the concentration range of the ternary system are calculated using the Novotny model [26] and correlated using the NRTL model derived by Humffray [33]. From this correlation, the pressure derivatives of the NRTL model parameters are obtained. It is important to mention that the NRTL binary interaction energy parameters at different temperatures are not available in the literature; only the values at 298.15 K have been reported by Chen et al. [39]. However, Morales et al. [40] reported the mean ionic activity coefficient data at 288.15, 298.15 and 308.15 K. These data were correlated using the NRTL model [39], where the temperature dependence of the NRTL binary interaction energy parameters is defined by ji = aji +

(25)

The values obtained for the aca,w , bca,w , aw,ca and bw,ca parameters were −4.8812, 10266.8565, −9.3810 and 437.8999, respectively (overall absolute average deviation = 1.1%). For this system, the non-randomness factor, ˛, is fixed at a value of 0.2, as this value

1.35

a

The standard uncertainties u are u(wi ) = 0.01%, u(T) = 0.01 K for density, u(T) = 0.02 K for viscosity, u() = 5 × 10−5 g cm−3 , u( ) = 4 × 10−2 mPa s for

> 32.67 mPa s and 1.5 × 10−3 mPa s for < 32.67 mPa s.

-3

1.3

ρ/g•cm

of experimental data, a, b, c and d, and are presented in Table 2. As observed in Fig. 1, each set of experimental data (a, b, c and d) follows the tendency of the binodal curve. In this figure is also included the binodal curve (solid line). For all of the mixtures studied, both density and viscosity decreased with increasing temperature, as shown in Table 2. For a given temperature, the dependence of these two properties on mass concentration can be observed at fixed concentrations of NaClO4 or PEG-4000. Both density and viscosity increase with increasing concentrations of NaClO4 or PEG-4000. This behavior is observed for each of the three temperatures studied in this work. The effect of the temperature on density and viscosity properties at different concentrations of NaClO4 and PEG-4000 is shown in Figs. 2 and 3.

bji RT

a

1.25

b

1.2

1.15

1.1 40

c

d 30 20

NaClO4(w/w%)

10

0

10

20

30

40

PEG(w/w%)

Fig. 2. Density of the NaClO4 + PEG 4000 + H2 O ternary system at different temperatures, following the binodal curve tendency: , 288.15 K; , 298.15 K; ×, 308.15 K; and — calculated. a, b, c and d are the sets of concentrations prepared, cf. Table 2.

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

of the molar volume of PEG 4000 and the NRTL binary interaction energy parameters at different temperatures. The molar volume value of 3325 cm3 mol−1 at 288.15 K was taken from Graber et al. [11]. The values at other temperatures were calculated from the infinite dilution apparent molar volume of polymer at the defined temperature. These data were correlated from the density data of aqueous solutions of PEG [10] (V2 = 3359 and 3385 cm3 mol−1 at 298.15 and 308.15 K, respectively). The NRTL binary interaction energy parameters were calculated from the correlation of the solvent activity data [28,29] by Chen’s model for polymer solutions [23]. The temperature dependence of the NRTL binary interaction energy parameters as recommended by Chen and Evans [22] was defined as

200

η/mPa•s

150 a 100

b

50

c d

0 30

 =a+b 30

0

has no significant impact on the behavior of the model in the range of 0.2–0.3. Additionally, the value of Debye–Hückel constant, Av , is obtained from Krumgalz et al. [41]. The parameters were evaluated using the following objective function (OF): n  (exp − calc )2

(26)

exp

This OF guaranteed short computing times and suitable exactitude for representing the experimental data with respect to the correlated data. The fitting parameters of the pressure derivatives of the NRTL model parameters and the infinite dilution apparent molar volume of the electrolyte at different temperatures are presented in Table 3. The temperature dependence for these variables is defined by Eqs. (4) and (5), respectively. The density data for the PEG 4000 + H2 O system were taken from Graber et al. [11] and Zafarani-Moattar and Mehrdad [10]. Correlation of the density data using the NRTL model [34] requires values Table 3 v and avw,s , and ternary NRTL parameters, Binary NRTL parameters, avca,w , avw,ca , as,w as,ca , aca,s , avs,ca and avca,s , for density. T/K

288.15 298.15 308.15 Overall 288.15 298.15 308.15 Overall 288.15 298.15 308.15 Overall

av

ca,w

1 298.15



+c

 298.15 − T T

+ ln

T 298.15



(27)

for  sw a = 1.0786, b = −1.4250, c = 22.4747 for  ws a = −0.5172, b = −1.0023, c = 16.9638

PEG(w/w%)

Fig. 3. Viscosity of the NaClO4 + PEG 4000 + H2 O ternary system at different temperatures, following the binodal curve tendency: , 288.15 K; , 298.15 K; ×, 308.15 K; and — calculated. a, b, c and d are the sets of concentrations prepared, cf. Table 2.

1=1

T

−

10 10

NaClO 4 (w/w%)

OF =

1

The coefficients obtained by least squares were

20

20

27

av

V20 /cm3

w,ca

0.7074

20.4701

avs,w −0.0862

avw,s −0.0027

as,ca −0.0797

aca,s 0.0058

mol

−1

a

b

6.2952

0.1278

AAD [%]

0.19 0.48 0.71 0.46 0.06 0.08 0.07 0.07

avs,ca 8.2844



Average absolute deviation (AAD) = (1/Np)

avca,s −0.7479

0.05 0.02 0.03 0.03

[(exp − cal)/exp] × 100%.

These values are valid only from 298.15 to 328.15 K as solvent activity data at 288.15 K have not been reported. The  sw and  ws values at 288.15 K were estimated using an extrapolation with the above coefficients. Finally, the Vex value (Eq. (6)) is calculated using the NRTL polymer model [23] derived by Sadeghi et al. [34], taking into consideration the unsymmetrical convention. For the interacv and  v , the temperature-dependent function tion parameters s,w w,s is defined by Eq. (4). The parameters were also evaluated using Eq. (26); these are presented in Table 3. In the correlations of the solvent activity and density data, the ˛ values were fixed at 0.2. As observed in Table 3, the absolute average deviation (AAD) values for the binary aqueous electrolyte system are acceptable, considering that only four parameters were required to correlate the experimental data at the three temperatures. The highest AAD value was obtained at 308.15 K, and the lowest AAD was obtained at 288.15 K. This deviation increased with increasing salt concentration at all three temperatures. The AAD values for the aqueous polymer system, also presented in Table 3, show good agreement between experimental and correlated values; this agreement is even better than the AAD values of the binary aqueous electrolyte system and with two fewer adjustment parameters. The AAD values at the three temperatures show little variation between them. 4.3. Correlation of density data of ternary system v , and The four new binary interaction parameters,  s,ca ,  ca,s , s,ca v , for the segment–salt interaction pair and salt–segment interca,s action pair are determined by fitting the ternary system density data using Eqs. (10) and (22) and using the binary parameters determined previously. The temperature dependence is expressed by Eq. (4). The non-randomness parameter, ˛, was also fixed at 0.2. The objective function, defined by Eq. (26), and the parameters are also presented in Table 3. To demonstrate the reliability of the model, a comparison between the experimental and correlated values is shown in Fig. 2. As the figure shows, there is good agreement between the experimental and calculated results. For the ternary system at the three temperatures, the AAD values, presented in Table 3, indicate that the adjustment with only four parameters more is of good quality; the adjustment is even better than the AAD values of the constituent binary systems. As can also be observed, the AAD values of the ternary system are not directly related to the AAD values of the constituent binary

28

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

Table 4 Binary NRTL parameters, aca,w , aw,ca , asw and aws , and ternary NRTL parameters, as,ca and aca,s , for viscosity. T/K

aca,w /J mol−1

aw,ca /J mol−1

288.15 298.15 308.15 Overall

−30,753.2802

219,828.7640

asw /J mol−1 −13,311.3013

aws /J mol−1 16,772.4405

as,ca /J mol−1 21,721.8925

aca,s /J mol−1 112,103.5333

288.15 298.15 308.15 Overall 288.15 298.15 308.15 Overall



Absolute average deviation (AAD) = (1/Np)

AAD [%] – 4.76 5.06 6.03 – 8.46 7.66 6.92 24.76 22.44 24.81 24.00

[( exp − cal)/ exp] × 100%.

systems. For the ternary system, the AAD values at 288.15 K are slightly higher than the other AAD values at the other temperatures. 4.4. Correlation of the viscosity data of binary systems In the present work, viscosity values of the NaClO4 + H2 O binary system reported by Janssen [31] were correlated using the NRTL model extended by Sadeghi [38], where the temperature dependence of the interaction parameters,  ca,w and  w,ca , is defined by Eq. (9). As viscosity data at 288.15 K were not available, the interaction parameters determined at the others temperatures (from 298.15 to 338.15 K) were used to extrapolate the  ca,w and  w,ca values at 288.15 K. The interaction parameters of the binary aqueous salt system are presented in Table 4. The empirical parameter, A, has the following temperature dependence: A2 = A0 + A1 (T − 298.15)

(28)

where A0 and A1 are adjustable parameters determined to be −0.1482 and 0.0040, respectively. The A2 value at 288.15 K was extrapolated from the adjustable parameter values mentioned above. The model parameters were estimated by minimizing the following objective function: OF =

n  ( exp − calc )2 1=1

exp

(29)

This OF also guaranteed short computing times and suitable exactitude for representing the experimental data with respect to the correlated data. For the PEG 4000 + H2 O system, the viscosity data reported by Silva et al. [32] were correlated using Eq. (23). The temperature dependence of the interaction parameters,  s,w and  w,s , is defined by Eq. (9). The previous methodology was used with adjustments for the lack of viscosity data at 288.15 K. As these viscosity data (at 288.15 K) have not been reported, the interaction parameter value was interpolated at this temperature. The interaction parameters of the binary aqueous polymer system are shown in Table 4. The adjustable parameter, A3 , has the same temperature dependence, as mentioned above, and the A0 and A1 values were determined to be 98.7306 and −0.1271, respectively. Additionally, the A3 value at 288.15 K was interpolated. The interaction parameters were evaluated using Eq. (29). As observed in Table 4, the binary parameters aca,w and as,w have negative values, and the aw,ca and aw,s parameters have positive values. From Eq. (9), the electrolyte–solvent and segment–solvent parameters, aca,w and as,w , are the differences of the

interaction energies between the ion–solvent molecule pair and the solvent–solvent pair and between the segment–solvent molecule pair and the solvent–solvent pair, respectively. Moreover, the solvent–electrolyte and solvent–segment parameters, aw,ca and aw,s , are the differences of the interaction energies between the solvent–ion pair and the cation–anion pair and between the solvent–segment pair and the segment–segment pair, respectively. The negative values of aca,w and as,w indicate that for the activated state, the interaction between the solvent and the ionic species and between the solvent and the segment, respectively, are stronger than the interactions between two solvent molecules. The positive values of the aw,ca and aw,s parameters indicate that the interactions between the cation and the anion and between two segments are stronger than the interactions between the solvent molecule and the ionic species and between the solvent molecule and the segment, respectively. The AAD of the aqueous electrolyte system increases as the salt concentration increases, with high deviations at the sodium perchlorate mass fractions of 0.3–0.375. The AAD values of the aqueous polymer system are similar to the AAD values of the aqueous electrolyte system. Unlike the deviations observed with increasing salt concentrations, increasing polymer concentrations did not increase the deviations, making it impossible to identify a polymer concentration range with high deviations. It is possible that parts of the deviations are within experimental uncertainties. The overall AADs of the binary systems, presented in Table 4, are the AADs of the whole temperature range of the experimental data reported in the literature [31,32]. 4.5. Correlation of viscosity data of ternary system The methodology and equations described above were applied to this system. It is important to mention that the A parameter is assumed to be equal to A2 + A3 + A4 , where A2 and A3 are adjustable parameters obtained from correlation of the viscosity values for the binary aqueous solutions and the temperature dependence of A4 parameter is defined by Eq. (28). The A0 and A1 values were determined to be −87.6375 and 0.4366, respectively. Unlike the density data analysis where the value of the ˛ parameter was fixed at 0.2, ˛ was considered to be an adjustment parameter in the correlation of the viscosity data, as better results were obtained by fixing it at 0.1596 for all systems. The function objective is defined above. The parameters are presented in Table 4. A comparison between the experimental and correlated values is shown in Fig. 3; there is good agreement between the experimental and calculated results. It should be noted that high deviations were also observed at high concentrations, particularly for the data set at 288.15 K. These deviations could be attributed to the extrapolated and interpolated values of the interaction parameters of the binary systems at 288.15 K and used in the correlation of the experimental data of the ternary system. At the experimental temperatures, the AADs of the ternary system, also reported in Table 4, are higher than the AADs of the binary systems. This difference could be due to the complexity of the system and the molecular interactions at high concentrations of poly(ethylene glycol). The positive values of the as,ca and aca,s parameters presented in Table 4 indicate that the interactions between the cation and the anion and between two segments are stronger than the interactions between the segment and the ionic species and between the ionic species and the segment, respectively. 5. Conclusions This work provides density and viscosity data for the NaClO4 + PEG 4000 + H2 O ternary system at 288.15, 298.15 and 308.15 K, which are needed for the engineering calculations

Y.P. Jimenez et al. / Fluid Phase Equilibria 334 (2012) 22–29

used in process design. The experimentally determined values for density and viscosity decreased with increasing temperature. These experimental data for the ternary system were successfully correlated with the NRTL model using parameters from the binary contributions. Our work on the ternary system also indicated that the interactions between the cation and the anion and between two segments are stronger than the interactions between the segment and the ionic species. These interactions decreased as the temperature increased. In the modeling work, some data points were estimated due to a lack of experimental data. As these values had little effect on the final model predictions, they are considered to be appropriate for engineering calculations. List of symbols

Ai Av A ci d e gex Ix K m M1 M2 NA Nsolu p R T Vex V10 V20 xi Zi

empirical adjustable parameter (L mol−1 ) Debye–Hückel constant for volume Debye–Hückel constant for osmotic coefficient molar concentration of the solute species i density of solution electronic charge molar excess Gibbs energy ionic strength in mole fraction scale Boltsmann constant molality molar mass of solvent (kg mol−1 ) molar mass of electrolyte or polymer (kg mol−1 ) Avogadro’s number total number of solute species pressure gas constant absolute temperature excess volume molar volume of pure solvent infinite dilution apparent molar volume of electrolyte or polymer mole fraction of species i absolute charge of ion i

Greek symbols ˛ non-randomness factor in the NRTL model i activity coefficient of species i permitivity of vaccum ε0 εT dielectric constant of pure solvent  the pi number closest distance parameter  ϕ segment fraction NRTL binary interaction energy parameter  ijv pressure derivatives of  ij Subscripts a anion c cation ca salt any species, segment, ions and solvent i, j, k I, J any species, polymer, ions and solvent m molecular species p polymer s segment w water calc calculated value experimental value exp LR long-range SR short-range

29

Superscripts combinatorial Comb ex excess property long-range LR SR short-range PDH Pitzer–Debye–Hückel non-random two liquid NRTL * unsymmetric convention Acknowledgments The authors thank CONICYT-Chile for the support provided through Fondecyt Project No. 1070909 and CICITEM (Project R10C1004), to Universidad de Antofagasta by the publication scholarship, and Y.J.B thanks also to CONICYT-Chile by the doctoral scholarship. References [1] B.A. Andrews, J.A. Asenjo, In aqueous two-phase partitioning, in: E. Harris, S. Angal (Eds.), Protein Purification Methods: A Practical Approach, IRL Press, Oxford, 1989. [2] J.C. Merchuk, B.A. Andrews, J.A. Asenjo, J. Chromatogr. B: Biomed. Sci. Appl. 711 (1998) 285–293. [3] T.A. Graber, B.A. Andrews, J.A. Asenjo, J. Chromatogr. B 743 (2000) 57–64. [4] R.D. Rogers, A.H. Bond, C.B. Bauer, J. Zhang, S.T. Griffin, J. Chromatogr. B: Biomed. Appl. 680 (1996) 221–229. [5] S.K. Spear, S.T. Griffin, J.G. Huddleston, Ind. Eng. Chem. Res. 39 (2000) 3173–3180. [6] M.E. Taboada, T.A. Graber, B.A. Andrews, J.A. Asenjo, J. Chromatogr. B 743 (2000) 101–105. [7] S.M. Snyder, K.D. Cole, D.C. Sziag, J. Chem. Eng. Data 37 (1992) 268–274. [8] L.H. Mei, D.Q. Lin, Z.Q. Zhu, Z.H. Han, J. Chem. Eng. Data 40 (1995) 1168–1171. [9] M.T. Zafarani-Moattar, A. Salabat, M. Kabiri-Badr, J. Chem. Eng. Data 40 (1995) 559–562. [10] M.T. Zafarani-Moattar, A. Mehrdad, J. Chem. Eng. Data 45 (2000) 386–390. [11] T.A. Graber, H.R. Galleguillos, J.A. Asenjo, B.A. Andrews, J. Chem. Eng. Data 47 (2002) 174–178. [12] T.A. Graber, H.R. Galleguillos, C. Cespedes, M.E. Taboada, J. Chem. Eng. Data 49 (2004) 1254–1257. [13] J.T. Telis-Romero, J.R. Coimbra, A.I. Gabas, E.E.G. Rojas, L.A. Minim, V.R.N. Telis, J. Chem. Eng. Data 49 (2004) 1340–1343. [14] C.B. Gonc¸alves, N. Trevisan Jr., A.J.A. Meirelles, J. Chem. Eng. Data 50 (2005) 177–181. [15] M.E. Taboada, H.R. Galleguillos, T.A. Graber, J. Chem. Eng. Data 50 (2005) 264–269. [16] M.T. Zafarani-Moattar, Sh. Hamzehzadeh, J. Chem. Eng. Data 50 (2005) 603–607. [17] T. Murugesan, M. Perumalsamy, J. Chem. Eng. Data 50 (2005) 1290–1293. [18] M. Perumalsamy, T. Murugesan, J. Chem. Eng. Data 54 (2009) 1359–1366. [19] I. Regupathi, S. Murugesan, S.P. Amaresh, R. Govindarajan, M. Thanabalan, J. Chem. Eng. Data 54 (2009) 1100–1106. [20] R. Sadeghi, R. Golabiazar, M. Ziaii, J. Chem. Eng. Data 55 (2010) 125–133. [21] Y.P. Jimenez, H.R. Galleguillos, J. Chem. Thermodyn. 42 (2010) 419–424. [22] C.C. Chen, L.B. Evans, AIChE J. 32 (1986) 444–454. [23] C.C. Chen, Fluid Phase Equilibr. 83 (1993) 301–312. [24] C.C. Chen, Y. Song, AIChE J. 50 (2004) 1928–1941. [25] M.T. Zafarani-Moattar, R. Sadeghi, Fluid Phase Equilibr. 203 (2002) 177–191. [26] P. Novotny, O. Sohnel, J. Chem. Eng. Data 33 (1988) 49–55. [27] S. Kirincic, C. Klofutar, Fluid Phase Equilibr. 149 (1998) 233–247. [28] D.Q. Lin, Z.Q. Zhu, L.H. Mei, L.R. Yang, J. Chem. Eng. Data 41 (1996) 1040–1042. [29] A. Eliassi, H. Modarress, G.A. Mansoori, J. Chem. Eng. Data 44 (1999) 52–55. [30] A. Eliassi, H. Modarress, G.A. Mansoori, J. Chem. Eng. Data 43 (1998) 719–721. [31] L.J.J. Janssen, J. Appl. Electrochem. 25 (1995) 291–293. [32] R.M.M. Silva, L.A. Minim, J.S.R. Coimbra, E.E. Garcia Rojas, L.H. Mendes da Silva, V.P. Rodrigues Minim, J. Chem. Eng. Data 52 (2007) 1567–1570. [33] A.A. Humffray, AIChE J. 35 (1989) 293–299. [34] R. Sadeghi, M.T. Zafarani-Moattar, A. Salabat, Ind. Eng. Chem. Res. 45 (2006) 2156–2162. [35] P.J. Flory, J. Chem. Phys. 9 (1941) 660–661. [36] K.S. Pitzer, J. Am. Chem. Soc. 102 (1980) 2902–2906. [37] M.T. Zafarani-Moattar, R. Majdan-Cegincara, Ind. Eng. Chem. Res. 48 (2009) 5833–5844. [38] R. Sadeghi, Fluid Phase Equilibr. 259 (2007) 157–164. [39] C.C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE J. 28 (1982) 588–596. [40] J.W. Morales, H.R. Galleguillos, F. Hernández-Luis, R. Rodríguez-Raposo, J. Chem. Eng. Data 56 (2011) 3449–3453. [41] B.S. Krumgalz, R. Pogorelskii, A. Sokolov, K.S. Pitzer, J. Phys. Chem. Ref. Data 29 (2000) 1123–1139.